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Key Concepts

Key Concepts

10.1 Finding Composite and Inverse Functions

  • Composition of Functions: The composition of functions ff and g,g, is written f∘gf∘g and is defined by
    (f∘g)(x)=f(g(x))(f∘g)(x)=f(g(x))

     We read f(g(x))f(g(x)) as ff of gg of x.x.
  • Horizontal Line Test: If every horizontal line, intersects the graph of a function in at most one point, it is a one-to-one function.
  • Inverse of a Function Defined by Ordered Pairs: If f(x)f(x) is a one-to-one function whose ordered pairs are of the form (x,y),(x,y), then its inverse function f−1(x)f−1(x) is the set of ordered pairs (y,x).(y,x).
  • Inverse Functions: For every xx in the domain of one-to-one function ff and f−1,f−1,
    f−1(f(x))=xf(f−1(x))=xf−1(f(x))=xf(f−1(x))=x
  • How to Find the Inverse of a One-to-One Function:
    1. Step 1. Substitute y for f(x).f(x).
    2. Step 2. Interchange the variables x and y.
    3. Step 3. Solve for y.
    4. Step 4. Substitute f−1(x)f−1(x) for y.y.
    5. Step 5. Verify that the functions are inverses.

10.2 Evaluate and Graph Exponential Functions

  • Properties of the Graph of f(x)=ax:f(x)=ax:
    when a>1a>1 when 0<a<10<a<1
    Domain (−∞,∞)(−∞,∞) Domain (−∞,∞)(−∞,∞)
    Range (0,∞)(0,∞) Range (0,∞)(0,∞)
    xx-intercept none xx-intercept none
    yy-intercept (0,1)(0,1) yy-intercept (0,1)(0,1)
    Contains (1,a),(−1,1a)(1,a),(−1,1a) Contains (1,a),(−1,1a)(1,a),(−1,1a)
    Asymptote xx-axis, the line y=0y=0 Asymptote xx-axis, the line y=0y=0
    Basic shape increasing Basic shape decreasing

    This figure has two parts. On the left, we have a curve that passes through (negative 1, 1 over a) through (0, 1) to (1, a). On the right, where a is noted to be less than 1, we have a curve that passes through (negative 1, 1 over a) through (0, 1) to (1, a).
  • One-to-One Property of Exponential Functions:
     For a>0a>0 and a≠1,a≠1,
    Ifax=ay,thenx=y.Ifax=ay,thenx=y.
  • How to Solve an Exponential Equation
    1. Step 1. Write both sides of the equation with the same base, if possible.
    2. Step 2. Write a new equation by setting the exponents equal.
    3. Step 3. Solve the equation.
    4. Step 4. Check the solution.
  • Compound Interest: For a principal, P,P, invested at an interest rate, r,r, for tt years, the new balance, A,A, is
     A=P(1+rn)ntwhen compoundedntimes a year. A=Pertwhen compounded continuously. A=P(1+rn)ntwhen compoundedntimes a year. A=Pertwhen compounded continuously.
  • Exponential Growth and Decay: For an original amount, A0A0 that grows or decays at a rate, r,r, for a certain time t,t, the final amount,A,A, is A=A0ert.A=A0ert.

10.3 Evaluate and Graph Logarithmic Functions

  • Properties of the Graph of y=logax:y=logax:
    when a>1a>1 when 0<a<10<a<1
    Domain (0,∞)(0,∞) Domain (0,∞)(0,∞)
    Range (−∞,∞)(−∞,∞) Range (−∞,∞)(−∞,∞)
    x-intercept (1,0)(1,0) x-intercept (1,0)(1,0)
    y-intercept none y-intercept none
    Contains (a,1),(a,1),(1a,−1)(1a,−1) Contains (a,1),(a,1),(1a,−1)(1a,−1)
    Asymptote y-axis Asymptote y-axis
    Basic shape increasing Basic shape decreasing

    This figure shows that, for a greater than 1, the logarithmic curve going through the points (1 over a, negative 1), (1, 0), and (a, 1). This figure shows that, for a greater than 0 and less than 1, the logarithmic curve going through the points (a, 1), (1, 0), and (1 over a, negative 1).
  • Decibel Level of Sound: The loudness level, DD, measured in decibels, of a sound of intensity, II, measured in watts per square inch is D=10log(I10−12).D=10log(I10−12).
  • Earthquake Intensity: The magnitude RR of an earthquake is measured by R=logI,R=logI, where II is the intensity of its shock wave.

10.4 Use the Properties of Logarithms

  • Properties of Logarithms
    loga1=0logaa=1loga1=0logaa=1
  • Inverse Properties of Logarithms
    • For a>0,a>0,x>0x>0 and a≠1a≠1
      alogax=xlogaax=xalogax=xlogaax=x
  • Product Property of Logarithms
    • If M>0,N>0,a>0M>0,N>0,a>0 and a≠1,a≠1, then,
      logaM·N=logaM+logaNlogaM·N=logaM+logaN

       The logarithm of a product is the sum of the logarithms.
  • Quotient Property of Logarithms
    • If M>0,N>0,a>0M>0,N>0,a>0 and a≠1,a≠1, then,
      logaMN=logaM−logaNlogaMN=logaM−logaN

       The logarithm of a quotient is the difference of the logarithms.
  • Power Property of Logarithms
    • If M>0,a>0,a≠1M>0,a>0,a≠1 and pp is any real number then,
      logaMp=plogaMlogaMp=plogaM

       The log of a number raised to a power is the product of the power times the log of the number.
  • Properties of Logarithms Summary
     If M>0,a>0,a≠1M>0,a>0,a≠1 and pp is any real number then,
    Property Base aa Base ee
    loga1=0loga1=0 ln1=0ln1=0
    logaa=1logaa=1 lne=1lne=1
    Inverse Properties alogax=x logaax=xalogax=x logaax=x elnx=x lnex=xelnx=x lnex=x
    Product Property of Logarithms loga(M·N)=logaM+logaNloga(M·N)=logaM+logaN ln(M·N)=lnM+lnNln(M·N)=lnM+lnN
    Quotient Property of Logarithms logaMN=logaM−logaNlogaMN=logaM−logaN lnMN=lnM−lnNlnMN=lnM−lnN
    Power Property of Logarithms logaMp=plogaMlogaMp=plogaM lnMp=plnMlnMp=plnM
  • Change-of-Base Formula
     For any logarithmic bases a and b, and M>0,M>0,
    logaM=logbMlogbalogaM=logMlogalogaM=lnMlna new basebnew base 10new baseelogaM=logbMlogbalogaM=logMlogalogaM=lnMlna new basebnew base 10new basee

10.5 Solve Exponential and Logarithmic Equations

  • One-to-One Property of Logarithmic Equations: For M>0,N>0,a>0,M>0,N>0,a>0, and a≠1a≠1 is any real number:
    IflogaM=logaN,thenM=N.IflogaM=logaN,thenM=N.
  • Compound Interest:
     For a principal, P, invested at an interest rate, r, for t years, the new balance, A, is:
    A=P(1+rn)ntwhen compoundedntimes a year. A=Pertwhen compounded continuously.A=P(1+rn)ntwhen compoundedntimes a year. A=Pertwhen compounded continuously.
  • Exponential Growth and Decay: For an original amount, A0A0 that grows or decays at a rate, r, for a certain time t, the final amount, A, is A=A0ert.A=A0ert.